When trying to figure the probability of winning hands depending on whether you hit, stand, double, etc we commonly use infinite deck calculations. To help me with this I worked out a table (see below) which helps me calculate the probabilty of hitting to a specified number or more. For example if I'm on 12 and want to know what the chances of hitting to at least 18 I go to the bottom row of my table and find the odds of hitting to 6 greater than my current total (i.e. 18-12) and then multiplying it by 4 (i.e. 18, 19, 20 & 21). The answer is 11.14% x 4 = 44.57%. This is different from the probabilty of hitting to 18 or above in one hit only (4 x 1/13 = 30.8%) as it assumes you hit until you get to 18 or more or bust trying. I have encountered problems using it though when the total I want to hit to is or could be 10 greater than my current total or is greater than 11. Say for example I'm on 8 and want to know the chances of hitting to at least 18 which is 10 greater than my current total, according to my table it should be 38.06% x 4 (i.e. 18, 19, 20 & 21) = 152.3% which is of course impossible. So my question: what is an easier/better way of figuring these things out? Cheers Reachy
Probability of specified totals I can't figure out your method of calculating the probabilities of specific totals except perhaps you overlook that an Ace can be an 11 as well as a 1. It looks like you are calculating the number of different combinations of cards that add up to a specified total. When hitting a starting total of 8 you can get 18 in 2 different ways with 2 cards if the cards are 8 & 2. The 1st hit card can be a 2 and the 2nd hit card can be an 8; or the 1st hit card can be the 8 and the 2nd hit card is the 2. However to get to total of 20 with 2 cards of 2 and 10 only one sequence can occur : the 1st hit cars is 2 and the 2nd hit card is 10. You cannot have the 1st hit card the 10 since the total is now 18 - the standing number so the 2 cannot be delt. In Peter Griffin's book "The Theory of BlackJack" he gives a 29 step basic program that calculates dealer probabilities when the dealer stands on all 17's. You have to modify the program to calculate dealer (and player) probabilities for any combination of hard and soft standing totals. Minor modifications are made for infinite deck. Below is the results I get for hitting 12 to 18 or bust and for hitting 8 to hard or soft 18 or bust for infinite deck. Starting total = 12.....................Starting total = 8 ...21...... .11142 ..........................21..... .07928 ...20...... .11142 ..........................20..... .07928 ...19...... .11142 ..........................19..... .13846 ...18...... .11142 ..........................18..... .36923 ...17...... .00000 ..........................17..... .00000 .stiff...... .00000 .........................stiff.... .00000 .bust..... .55430 .........................bust.... .33375 .....................BlueLight
